├── LICENSE ├── README.md ├── git-tips.txt └── matlab_code ├── bump.m ├── circleAndTorusIntervals.mat ├── demo.m ├── flexible_weighting_function.m ├── linear_ramp.m ├── make_PIs.m ├── make_PVs.m ├── minimal_example.m ├── rect_region.m ├── sixShapeClasses ├── ToyData_PD_TextFiles │ ├── README.md │ ├── ToyData_PD_n05_10_1_0.txt │ ├── ToyData_PD_n05_10_1_1.txt │ ├── ToyData_PD_n05_10_2_0.txt │ ├── ToyData_PD_n05_10_2_1.txt │ ├── ToyData_PD_n05_10_3_0.txt │ ├── ToyData_PD_n05_10_3_1.txt │ ├── ToyData_PD_n05_10_4_0.txt │ ├── ToyData_PD_n05_10_4_1.txt │ ├── ToyData_PD_n05_10_5_0.txt │ ├── ToyData_PD_n05_10_5_1.txt │ ├── ToyData_PD_n05_10_6_0.txt │ ├── ToyData_PD_n05_10_6_1.txt │ ├── ToyData_PD_n05_11_1_0.txt │ ├── ToyData_PD_n05_11_1_1.txt │ ├── ToyData_PD_n05_11_2_0.txt │ ├── ToyData_PD_n05_11_2_1.txt │ ├── ToyData_PD_n05_11_3_0.txt │ ├── ToyData_PD_n05_11_3_1.txt │ ├── ToyData_PD_n05_11_4_0.txt │ ├── ToyData_PD_n05_11_4_1.txt │ ├── ToyData_PD_n05_11_5_0.txt │ ├── ToyData_PD_n05_11_5_1.txt │ ├── ToyData_PD_n05_11_6_0.txt │ ├── ToyData_PD_n05_11_6_1.txt │ ├── ToyData_PD_n05_12_1_0.txt │ ├── ToyData_PD_n05_12_1_1.txt │ ├── ToyData_PD_n05_12_2_0.txt │ ├── ToyData_PD_n05_12_2_1.txt │ ├── ToyData_PD_n05_12_3_0.txt │ ├── ToyData_PD_n05_12_3_1.txt │ ├── ToyData_PD_n05_12_4_0.txt │ ├── ToyData_PD_n05_12_4_1.txt │ ├── ToyData_PD_n05_12_5_0.txt │ ├── ToyData_PD_n05_12_5_1.txt │ ├── ToyData_PD_n05_12_6_0.txt │ ├── ToyData_PD_n05_12_6_1.txt │ ├── ToyData_PD_n05_13_1_0.txt │ ├── ToyData_PD_n05_13_1_1.txt │ ├── ToyData_PD_n05_13_2_0.txt │ ├── ToyData_PD_n05_13_2_1.txt │ ├── ToyData_PD_n05_13_3_0.txt │ ├── ToyData_PD_n05_13_3_1.txt │ ├── ToyData_PD_n05_13_4_0.txt │ ├── ToyData_PD_n05_13_4_1.txt │ ├── ToyData_PD_n05_13_5_0.txt │ ├── ToyData_PD_n05_13_5_1.txt │ ├── ToyData_PD_n05_13_6_0.txt │ ├── ToyData_PD_n05_13_6_1.txt │ ├── ToyData_PD_n05_14_1_0.txt │ ├── ToyData_PD_n05_14_1_1.txt │ ├── ToyData_PD_n05_14_2_0.txt │ ├── ToyData_PD_n05_14_2_1.txt │ ├── ToyData_PD_n05_14_3_0.txt │ ├── ToyData_PD_n05_14_3_1.txt │ ├── ToyData_PD_n05_14_4_0.txt │ ├── ToyData_PD_n05_14_4_1.txt │ ├── ToyData_PD_n05_14_5_0.txt │ ├── ToyData_PD_n05_14_5_1.txt │ ├── ToyData_PD_n05_14_6_0.txt │ ├── ToyData_PD_n05_14_6_1.txt │ ├── ToyData_PD_n05_15_1_0.txt │ ├── ToyData_PD_n05_15_1_1.txt │ ├── ToyData_PD_n05_15_2_0.txt │ ├── ToyData_PD_n05_15_2_1.txt │ ├── ToyData_PD_n05_15_3_0.txt │ ├── ToyData_PD_n05_15_3_1.txt │ ├── ToyData_PD_n05_15_4_0.txt │ ├── ToyData_PD_n05_15_4_1.txt │ ├── ToyData_PD_n05_15_5_0.txt │ ├── ToyData_PD_n05_15_5_1.txt │ ├── ToyData_PD_n05_15_6_0.txt │ ├── ToyData_PD_n05_15_6_1.txt │ ├── ToyData_PD_n05_16_1_0.txt │ ├── ToyData_PD_n05_16_1_1.txt │ ├── ToyData_PD_n05_16_2_0.txt │ ├── ToyData_PD_n05_16_2_1.txt │ ├── ToyData_PD_n05_16_3_0.txt │ ├── ToyData_PD_n05_16_3_1.txt │ ├── ToyData_PD_n05_16_4_0.txt │ ├── ToyData_PD_n05_16_4_1.txt │ ├── ToyData_PD_n05_16_5_0.txt │ ├── ToyData_PD_n05_16_5_1.txt │ ├── ToyData_PD_n05_16_6_0.txt │ ├── ToyData_PD_n05_16_6_1.txt │ ├── ToyData_PD_n05_17_1_0.txt │ ├── ToyData_PD_n05_17_1_1.txt │ ├── ToyData_PD_n05_17_2_0.txt │ ├── ToyData_PD_n05_17_2_1.txt │ ├── ToyData_PD_n05_17_3_0.txt │ ├── ToyData_PD_n05_17_3_1.txt │ ├── ToyData_PD_n05_17_4_0.txt │ ├── ToyData_PD_n05_17_4_1.txt │ ├── ToyData_PD_n05_17_5_0.txt │ ├── ToyData_PD_n05_17_5_1.txt │ ├── ToyData_PD_n05_17_6_0.txt │ ├── ToyData_PD_n05_17_6_1.txt │ ├── ToyData_PD_n05_18_1_0.txt │ ├── ToyData_PD_n05_18_1_1.txt │ ├── ToyData_PD_n05_18_2_0.txt │ ├── ToyData_PD_n05_18_2_1.txt │ ├── ToyData_PD_n05_18_3_0.txt │ ├── ToyData_PD_n05_18_3_1.txt │ ├── ToyData_PD_n05_18_4_0.txt │ ├── ToyData_PD_n05_18_4_1.txt │ ├── ToyData_PD_n05_18_5_0.txt │ ├── ToyData_PD_n05_18_5_1.txt │ ├── ToyData_PD_n05_18_6_0.txt │ ├── ToyData_PD_n05_18_6_1.txt │ ├── ToyData_PD_n05_19_1_0.txt │ ├── ToyData_PD_n05_19_1_1.txt │ ├── ToyData_PD_n05_19_2_0.txt │ ├── ToyData_PD_n05_19_2_1.txt │ ├── ToyData_PD_n05_19_3_0.txt │ ├── ToyData_PD_n05_19_3_1.txt │ ├── ToyData_PD_n05_19_4_0.txt │ ├── ToyData_PD_n05_19_4_1.txt │ ├── ToyData_PD_n05_19_5_0.txt │ ├── ToyData_PD_n05_19_5_1.txt │ ├── ToyData_PD_n05_19_6_0.txt │ ├── ToyData_PD_n05_19_6_1.txt │ ├── ToyData_PD_n05_1_1_0.txt │ ├── ToyData_PD_n05_1_1_1.txt │ ├── ToyData_PD_n05_1_2_0.txt │ ├── ToyData_PD_n05_1_2_1.txt │ ├── ToyData_PD_n05_1_3_0.txt │ ├── ToyData_PD_n05_1_3_1.txt │ ├── ToyData_PD_n05_1_4_0.txt │ ├── ToyData_PD_n05_1_4_1.txt │ ├── ToyData_PD_n05_1_5_0.txt │ ├── ToyData_PD_n05_1_5_1.txt │ ├── ToyData_PD_n05_1_6_0.txt │ ├── ToyData_PD_n05_1_6_1.txt │ ├── ToyData_PD_n05_20_1_0.txt │ ├── ToyData_PD_n05_20_1_1.txt │ ├── ToyData_PD_n05_20_2_0.txt │ ├── ToyData_PD_n05_20_2_1.txt │ ├── ToyData_PD_n05_20_3_0.txt │ ├── ToyData_PD_n05_20_3_1.txt │ ├── ToyData_PD_n05_20_4_0.txt │ ├── ToyData_PD_n05_20_4_1.txt │ ├── ToyData_PD_n05_20_5_0.txt │ ├── ToyData_PD_n05_20_5_1.txt │ ├── ToyData_PD_n05_20_6_0.txt │ ├── ToyData_PD_n05_20_6_1.txt │ ├── ToyData_PD_n05_21_1_0.txt │ ├── ToyData_PD_n05_21_1_1.txt │ ├── ToyData_PD_n05_21_2_0.txt │ ├── ToyData_PD_n05_21_2_1.txt │ ├── ToyData_PD_n05_21_3_0.txt │ ├── ToyData_PD_n05_21_3_1.txt │ ├── ToyData_PD_n05_21_4_0.txt │ ├── ToyData_PD_n05_21_4_1.txt │ ├── ToyData_PD_n05_21_5_0.txt │ ├── ToyData_PD_n05_21_5_1.txt │ ├── ToyData_PD_n05_21_6_0.txt │ ├── ToyData_PD_n05_21_6_1.txt │ ├── ToyData_PD_n05_22_1_0.txt │ ├── ToyData_PD_n05_22_1_1.txt │ ├── ToyData_PD_n05_22_2_0.txt │ ├── ToyData_PD_n05_22_2_1.txt │ ├── ToyData_PD_n05_22_3_0.txt │ ├── ToyData_PD_n05_22_3_1.txt │ ├── ToyData_PD_n05_22_4_0.txt │ ├── ToyData_PD_n05_22_4_1.txt │ ├── ToyData_PD_n05_22_5_0.txt │ ├── ToyData_PD_n05_22_5_1.txt │ ├── ToyData_PD_n05_22_6_0.txt │ ├── ToyData_PD_n05_22_6_1.txt │ ├── ToyData_PD_n05_23_1_0.txt │ ├── ToyData_PD_n05_23_1_1.txt │ ├── ToyData_PD_n05_23_2_0.txt │ ├── ToyData_PD_n05_23_2_1.txt │ ├── ToyData_PD_n05_23_3_0.txt │ ├── ToyData_PD_n05_23_3_1.txt │ ├── ToyData_PD_n05_23_4_0.txt │ ├── ToyData_PD_n05_23_4_1.txt │ ├── ToyData_PD_n05_23_5_0.txt │ ├── ToyData_PD_n05_23_5_1.txt │ ├── ToyData_PD_n05_23_6_0.txt │ ├── ToyData_PD_n05_23_6_1.txt │ ├── ToyData_PD_n05_24_1_0.txt │ ├── ToyData_PD_n05_24_1_1.txt │ ├── ToyData_PD_n05_24_2_0.txt │ ├── ToyData_PD_n05_24_2_1.txt │ ├── ToyData_PD_n05_24_3_0.txt │ ├── ToyData_PD_n05_24_3_1.txt │ ├── ToyData_PD_n05_24_4_0.txt │ ├── ToyData_PD_n05_24_4_1.txt │ ├── ToyData_PD_n05_24_5_0.txt │ ├── ToyData_PD_n05_24_5_1.txt │ ├── ToyData_PD_n05_24_6_0.txt │ ├── ToyData_PD_n05_24_6_1.txt │ ├── ToyData_PD_n05_25_1_0.txt │ ├── ToyData_PD_n05_25_1_1.txt │ ├── ToyData_PD_n05_25_2_0.txt │ ├── ToyData_PD_n05_25_2_1.txt │ ├── ToyData_PD_n05_25_3_0.txt │ ├── ToyData_PD_n05_25_3_1.txt │ ├── ToyData_PD_n05_25_4_0.txt │ ├── ToyData_PD_n05_25_4_1.txt │ ├── ToyData_PD_n05_25_5_0.txt │ ├── ToyData_PD_n05_25_5_1.txt │ ├── ToyData_PD_n05_25_6_0.txt │ ├── ToyData_PD_n05_25_6_1.txt │ ├── ToyData_PD_n05_2_1_0.txt │ ├── ToyData_PD_n05_2_1_1.txt │ ├── ToyData_PD_n05_2_2_0.txt │ ├── ToyData_PD_n05_2_2_1.txt │ ├── ToyData_PD_n05_2_3_0.txt │ ├── ToyData_PD_n05_2_3_1.txt │ ├── ToyData_PD_n05_2_4_0.txt │ ├── ToyData_PD_n05_2_4_1.txt │ ├── ToyData_PD_n05_2_5_0.txt │ ├── ToyData_PD_n05_2_5_1.txt │ ├── ToyData_PD_n05_2_6_0.txt │ ├── ToyData_PD_n05_2_6_1.txt │ ├── ToyData_PD_n05_3_1_0.txt │ ├── ToyData_PD_n05_3_1_1.txt │ ├── ToyData_PD_n05_3_2_0.txt │ ├── ToyData_PD_n05_3_2_1.txt │ ├── ToyData_PD_n05_3_3_0.txt │ ├── ToyData_PD_n05_3_3_1.txt │ ├── ToyData_PD_n05_3_4_0.txt │ ├── ToyData_PD_n05_3_4_1.txt │ ├── ToyData_PD_n05_3_5_0.txt │ ├── ToyData_PD_n05_3_5_1.txt │ ├── ToyData_PD_n05_3_6_0.txt │ ├── ToyData_PD_n05_3_6_1.txt │ ├── ToyData_PD_n05_4_1_0.txt │ ├── ToyData_PD_n05_4_1_1.txt │ ├── ToyData_PD_n05_4_2_0.txt │ ├── ToyData_PD_n05_4_2_1.txt │ ├── ToyData_PD_n05_4_3_0.txt │ ├── ToyData_PD_n05_4_3_1.txt │ ├── ToyData_PD_n05_4_4_0.txt │ ├── ToyData_PD_n05_4_4_1.txt │ ├── ToyData_PD_n05_4_5_0.txt │ ├── ToyData_PD_n05_4_5_1.txt │ ├── ToyData_PD_n05_4_6_0.txt │ ├── ToyData_PD_n05_4_6_1.txt │ ├── ToyData_PD_n05_5_1_0.txt │ ├── ToyData_PD_n05_5_1_1.txt │ ├── ToyData_PD_n05_5_2_0.txt │ ├── ToyData_PD_n05_5_2_1.txt │ ├── ToyData_PD_n05_5_3_0.txt │ ├── ToyData_PD_n05_5_3_1.txt │ ├── ToyData_PD_n05_5_4_0.txt │ ├── ToyData_PD_n05_5_4_1.txt │ ├── ToyData_PD_n05_5_5_0.txt │ ├── ToyData_PD_n05_5_5_1.txt │ ├── ToyData_PD_n05_5_6_0.txt │ ├── ToyData_PD_n05_5_6_1.txt │ ├── ToyData_PD_n05_6_1_0.txt │ ├── ToyData_PD_n05_6_1_1.txt │ ├── ToyData_PD_n05_6_2_0.txt │ ├── ToyData_PD_n05_6_2_1.txt │ ├── ToyData_PD_n05_6_3_0.txt │ ├── ToyData_PD_n05_6_3_1.txt │ ├── ToyData_PD_n05_6_4_0.txt │ ├── ToyData_PD_n05_6_4_1.txt │ ├── ToyData_PD_n05_6_5_0.txt │ ├── ToyData_PD_n05_6_5_1.txt │ ├── ToyData_PD_n05_6_6_0.txt │ ├── ToyData_PD_n05_6_6_1.txt │ ├── ToyData_PD_n05_7_1_0.txt │ ├── ToyData_PD_n05_7_1_1.txt │ ├── ToyData_PD_n05_7_2_0.txt │ ├── ToyData_PD_n05_7_2_1.txt │ ├── ToyData_PD_n05_7_3_0.txt │ ├── ToyData_PD_n05_7_3_1.txt │ ├── ToyData_PD_n05_7_4_0.txt │ ├── ToyData_PD_n05_7_4_1.txt │ ├── ToyData_PD_n05_7_5_0.txt │ ├── ToyData_PD_n05_7_5_1.txt │ ├── ToyData_PD_n05_7_6_0.txt │ ├── ToyData_PD_n05_7_6_1.txt │ ├── ToyData_PD_n05_8_1_0.txt │ ├── ToyData_PD_n05_8_1_1.txt │ ├── ToyData_PD_n05_8_2_0.txt │ ├── ToyData_PD_n05_8_2_1.txt │ ├── ToyData_PD_n05_8_3_0.txt │ ├── ToyData_PD_n05_8_3_1.txt │ ├── ToyData_PD_n05_8_4_0.txt │ ├── ToyData_PD_n05_8_4_1.txt │ ├── ToyData_PD_n05_8_5_0.txt │ ├── ToyData_PD_n05_8_5_1.txt │ ├── ToyData_PD_n05_8_6_0.txt │ ├── ToyData_PD_n05_8_6_1.txt │ ├── ToyData_PD_n05_9_1_0.txt │ ├── ToyData_PD_n05_9_1_1.txt │ ├── ToyData_PD_n05_9_2_0.txt │ ├── ToyData_PD_n05_9_2_1.txt │ ├── ToyData_PD_n05_9_3_0.txt │ ├── ToyData_PD_n05_9_3_1.txt │ ├── ToyData_PD_n05_9_4_0.txt │ ├── ToyData_PD_n05_9_4_1.txt │ ├── ToyData_PD_n05_9_5_0.txt │ ├── ToyData_PD_n05_9_5_1.txt │ ├── ToyData_PD_n05_9_6_0.txt │ ├── ToyData_PD_n05_9_6_1.txt │ ├── ToyData_PD_n1_10_1_0.txt │ ├── ToyData_PD_n1_10_1_1.txt │ ├── ToyData_PD_n1_10_2_0.txt │ ├── ToyData_PD_n1_10_2_1.txt │ ├── ToyData_PD_n1_10_3_0.txt │ ├── ToyData_PD_n1_10_3_1.txt │ ├── ToyData_PD_n1_10_4_0.txt │ ├── ToyData_PD_n1_10_4_1.txt │ ├── ToyData_PD_n1_10_5_0.txt │ ├── ToyData_PD_n1_10_5_1.txt │ ├── ToyData_PD_n1_10_6_0.txt │ ├── ToyData_PD_n1_10_6_1.txt │ ├── ToyData_PD_n1_11_1_0.txt │ ├── ToyData_PD_n1_11_1_1.txt │ ├── ToyData_PD_n1_11_2_0.txt │ ├── ToyData_PD_n1_11_2_1.txt │ ├── ToyData_PD_n1_11_3_0.txt │ ├── ToyData_PD_n1_11_3_1.txt │ ├── ToyData_PD_n1_11_4_0.txt │ ├── ToyData_PD_n1_11_4_1.txt │ ├── ToyData_PD_n1_11_5_0.txt │ ├── ToyData_PD_n1_11_5_1.txt │ ├── ToyData_PD_n1_11_6_0.txt │ ├── ToyData_PD_n1_11_6_1.txt │ ├── ToyData_PD_n1_12_1_0.txt │ ├── ToyData_PD_n1_12_1_1.txt │ ├── ToyData_PD_n1_12_2_0.txt │ ├── ToyData_PD_n1_12_2_1.txt │ ├── ToyData_PD_n1_12_3_0.txt │ ├── ToyData_PD_n1_12_3_1.txt │ ├── ToyData_PD_n1_12_4_0.txt │ ├── ToyData_PD_n1_12_4_1.txt │ ├── ToyData_PD_n1_12_5_0.txt │ ├── ToyData_PD_n1_12_5_1.txt │ ├── ToyData_PD_n1_12_6_0.txt │ ├── ToyData_PD_n1_12_6_1.txt │ ├── ToyData_PD_n1_13_1_0.txt │ ├── ToyData_PD_n1_13_1_1.txt │ ├── ToyData_PD_n1_13_2_0.txt │ ├── ToyData_PD_n1_13_2_1.txt │ ├── ToyData_PD_n1_13_3_0.txt │ ├── ToyData_PD_n1_13_3_1.txt │ ├── ToyData_PD_n1_13_4_0.txt │ ├── ToyData_PD_n1_13_4_1.txt │ ├── ToyData_PD_n1_13_5_0.txt │ ├── ToyData_PD_n1_13_5_1.txt │ ├── ToyData_PD_n1_13_6_0.txt │ ├── ToyData_PD_n1_13_6_1.txt │ ├── ToyData_PD_n1_14_1_0.txt │ ├── ToyData_PD_n1_14_1_1.txt │ ├── ToyData_PD_n1_14_2_0.txt │ ├── ToyData_PD_n1_14_2_1.txt │ ├── ToyData_PD_n1_14_3_0.txt │ ├── ToyData_PD_n1_14_3_1.txt │ ├── ToyData_PD_n1_14_4_0.txt │ ├── ToyData_PD_n1_14_4_1.txt │ ├── ToyData_PD_n1_14_5_0.txt │ ├── ToyData_PD_n1_14_5_1.txt │ ├── ToyData_PD_n1_14_6_0.txt │ ├── ToyData_PD_n1_14_6_1.txt │ ├── ToyData_PD_n1_15_1_0.txt │ ├── ToyData_PD_n1_15_1_1.txt │ ├── ToyData_PD_n1_15_2_0.txt │ ├── ToyData_PD_n1_15_2_1.txt │ ├── ToyData_PD_n1_15_3_0.txt │ ├── ToyData_PD_n1_15_3_1.txt │ ├── ToyData_PD_n1_15_4_0.txt │ ├── ToyData_PD_n1_15_4_1.txt │ ├── ToyData_PD_n1_15_5_0.txt │ ├── ToyData_PD_n1_15_5_1.txt │ ├── ToyData_PD_n1_15_6_0.txt │ ├── ToyData_PD_n1_15_6_1.txt │ ├── ToyData_PD_n1_16_1_0.txt │ ├── ToyData_PD_n1_16_1_1.txt │ ├── ToyData_PD_n1_16_2_0.txt │ ├── ToyData_PD_n1_16_2_1.txt │ ├── ToyData_PD_n1_16_3_0.txt │ ├── ToyData_PD_n1_16_3_1.txt │ ├── ToyData_PD_n1_16_4_0.txt │ ├── ToyData_PD_n1_16_4_1.txt │ ├── ToyData_PD_n1_16_5_0.txt │ ├── ToyData_PD_n1_16_5_1.txt │ ├── ToyData_PD_n1_16_6_0.txt │ ├── ToyData_PD_n1_16_6_1.txt │ ├── ToyData_PD_n1_17_1_0.txt │ ├── ToyData_PD_n1_17_1_1.txt │ ├── ToyData_PD_n1_17_2_0.txt │ ├── ToyData_PD_n1_17_2_1.txt │ ├── ToyData_PD_n1_17_3_0.txt │ ├── ToyData_PD_n1_17_3_1.txt │ ├── ToyData_PD_n1_17_4_0.txt │ ├── ToyData_PD_n1_17_4_1.txt │ ├── ToyData_PD_n1_17_5_0.txt │ ├── ToyData_PD_n1_17_5_1.txt │ ├── ToyData_PD_n1_17_6_0.txt │ ├── ToyData_PD_n1_17_6_1.txt │ ├── ToyData_PD_n1_18_1_0.txt │ ├── ToyData_PD_n1_18_1_1.txt │ ├── ToyData_PD_n1_18_2_0.txt │ ├── ToyData_PD_n1_18_2_1.txt │ ├── ToyData_PD_n1_18_3_0.txt │ ├── ToyData_PD_n1_18_3_1.txt │ ├── ToyData_PD_n1_18_4_0.txt │ ├── ToyData_PD_n1_18_4_1.txt │ ├── ToyData_PD_n1_18_5_0.txt │ ├── ToyData_PD_n1_18_5_1.txt │ ├── ToyData_PD_n1_18_6_0.txt │ ├── ToyData_PD_n1_18_6_1.txt │ ├── ToyData_PD_n1_19_1_0.txt │ ├── ToyData_PD_n1_19_1_1.txt │ ├── ToyData_PD_n1_19_2_0.txt │ ├── ToyData_PD_n1_19_2_1.txt │ ├── ToyData_PD_n1_19_3_0.txt │ ├── ToyData_PD_n1_19_3_1.txt │ ├── ToyData_PD_n1_19_4_0.txt │ ├── ToyData_PD_n1_19_4_1.txt │ ├── ToyData_PD_n1_19_5_0.txt │ ├── ToyData_PD_n1_19_5_1.txt │ ├── ToyData_PD_n1_19_6_0.txt │ ├── ToyData_PD_n1_19_6_1.txt │ ├── ToyData_PD_n1_1_1_0.txt │ ├── ToyData_PD_n1_1_1_1.txt │ ├── ToyData_PD_n1_1_2_0.txt │ ├── ToyData_PD_n1_1_2_1.txt │ ├── ToyData_PD_n1_1_3_0.txt │ ├── ToyData_PD_n1_1_3_1.txt │ ├── ToyData_PD_n1_1_4_0.txt │ ├── ToyData_PD_n1_1_4_1.txt │ ├── ToyData_PD_n1_1_5_0.txt │ ├── ToyData_PD_n1_1_5_1.txt │ ├── ToyData_PD_n1_1_6_0.txt │ ├── ToyData_PD_n1_1_6_1.txt │ ├── ToyData_PD_n1_20_1_0.txt │ ├── ToyData_PD_n1_20_1_1.txt │ ├── ToyData_PD_n1_20_2_0.txt │ ├── ToyData_PD_n1_20_2_1.txt │ ├── ToyData_PD_n1_20_3_0.txt │ ├── ToyData_PD_n1_20_3_1.txt │ ├── ToyData_PD_n1_20_4_0.txt │ ├── ToyData_PD_n1_20_4_1.txt │ ├── ToyData_PD_n1_20_5_0.txt │ ├── ToyData_PD_n1_20_5_1.txt │ ├── ToyData_PD_n1_20_6_0.txt │ ├── ToyData_PD_n1_20_6_1.txt │ ├── ToyData_PD_n1_21_1_0.txt │ ├── ToyData_PD_n1_21_1_1.txt │ ├── ToyData_PD_n1_21_2_0.txt │ ├── ToyData_PD_n1_21_2_1.txt │ ├── ToyData_PD_n1_21_3_0.txt │ ├── ToyData_PD_n1_21_3_1.txt │ ├── ToyData_PD_n1_21_4_0.txt │ ├── ToyData_PD_n1_21_4_1.txt │ ├── ToyData_PD_n1_21_5_0.txt │ ├── ToyData_PD_n1_21_5_1.txt │ ├── ToyData_PD_n1_21_6_0.txt │ ├── ToyData_PD_n1_21_6_1.txt │ ├── ToyData_PD_n1_22_1_0.txt │ ├── ToyData_PD_n1_22_1_1.txt │ ├── ToyData_PD_n1_22_2_0.txt │ ├── ToyData_PD_n1_22_2_1.txt │ ├── ToyData_PD_n1_22_3_0.txt │ ├── ToyData_PD_n1_22_3_1.txt │ ├── ToyData_PD_n1_22_4_0.txt │ ├── ToyData_PD_n1_22_4_1.txt │ ├── ToyData_PD_n1_22_5_0.txt │ ├── ToyData_PD_n1_22_5_1.txt │ ├── ToyData_PD_n1_22_6_0.txt │ ├── ToyData_PD_n1_22_6_1.txt │ ├── ToyData_PD_n1_23_1_0.txt │ ├── ToyData_PD_n1_23_1_1.txt │ ├── ToyData_PD_n1_23_2_0.txt │ ├── ToyData_PD_n1_23_2_1.txt │ ├── ToyData_PD_n1_23_3_0.txt │ ├── ToyData_PD_n1_23_3_1.txt │ ├── ToyData_PD_n1_23_4_0.txt │ ├── ToyData_PD_n1_23_4_1.txt │ ├── ToyData_PD_n1_23_5_0.txt │ ├── ToyData_PD_n1_23_5_1.txt │ ├── ToyData_PD_n1_23_6_0.txt │ ├── ToyData_PD_n1_23_6_1.txt │ ├── ToyData_PD_n1_24_1_0.txt │ ├── ToyData_PD_n1_24_1_1.txt │ ├── ToyData_PD_n1_24_2_0.txt │ ├── ToyData_PD_n1_24_2_1.txt │ ├── ToyData_PD_n1_24_3_0.txt │ ├── ToyData_PD_n1_24_3_1.txt │ ├── ToyData_PD_n1_24_4_0.txt │ ├── ToyData_PD_n1_24_4_1.txt │ ├── ToyData_PD_n1_24_5_0.txt │ ├── ToyData_PD_n1_24_5_1.txt │ ├── ToyData_PD_n1_24_6_0.txt │ ├── ToyData_PD_n1_24_6_1.txt │ ├── ToyData_PD_n1_25_1_0.txt │ ├── ToyData_PD_n1_25_1_1.txt │ ├── ToyData_PD_n1_25_2_0.txt │ ├── ToyData_PD_n1_25_2_1.txt │ ├── ToyData_PD_n1_25_3_0.txt │ ├── ToyData_PD_n1_25_3_1.txt │ ├── ToyData_PD_n1_25_4_0.txt │ ├── ToyData_PD_n1_25_4_1.txt │ ├── ToyData_PD_n1_25_5_0.txt │ ├── ToyData_PD_n1_25_5_1.txt │ ├── ToyData_PD_n1_25_6_0.txt │ ├── ToyData_PD_n1_25_6_1.txt │ ├── ToyData_PD_n1_2_1_0.txt │ ├── ToyData_PD_n1_2_1_1.txt │ ├── ToyData_PD_n1_2_2_0.txt │ ├── ToyData_PD_n1_2_2_1.txt │ ├── ToyData_PD_n1_2_3_0.txt │ ├── ToyData_PD_n1_2_3_1.txt │ ├── ToyData_PD_n1_2_4_0.txt │ ├── ToyData_PD_n1_2_4_1.txt │ ├── ToyData_PD_n1_2_5_0.txt │ ├── ToyData_PD_n1_2_5_1.txt │ ├── ToyData_PD_n1_2_6_0.txt │ ├── ToyData_PD_n1_2_6_1.txt │ ├── ToyData_PD_n1_3_1_0.txt │ ├── ToyData_PD_n1_3_1_1.txt │ ├── ToyData_PD_n1_3_2_0.txt │ ├── ToyData_PD_n1_3_2_1.txt │ ├── ToyData_PD_n1_3_3_0.txt │ ├── ToyData_PD_n1_3_3_1.txt │ ├── ToyData_PD_n1_3_4_0.txt │ ├── ToyData_PD_n1_3_4_1.txt │ ├── ToyData_PD_n1_3_5_0.txt │ ├── ToyData_PD_n1_3_5_1.txt │ ├── ToyData_PD_n1_3_6_0.txt │ ├── ToyData_PD_n1_3_6_1.txt │ ├── ToyData_PD_n1_4_1_0.txt │ ├── ToyData_PD_n1_4_1_1.txt │ ├── ToyData_PD_n1_4_2_0.txt │ ├── ToyData_PD_n1_4_2_1.txt │ ├── ToyData_PD_n1_4_3_0.txt │ ├── ToyData_PD_n1_4_3_1.txt │ ├── ToyData_PD_n1_4_4_0.txt │ ├── ToyData_PD_n1_4_4_1.txt │ ├── ToyData_PD_n1_4_5_0.txt │ ├── ToyData_PD_n1_4_5_1.txt │ ├── ToyData_PD_n1_4_6_0.txt │ ├── ToyData_PD_n1_4_6_1.txt │ ├── ToyData_PD_n1_5_1_0.txt │ ├── ToyData_PD_n1_5_1_1.txt │ ├── ToyData_PD_n1_5_2_0.txt │ ├── ToyData_PD_n1_5_2_1.txt │ ├── ToyData_PD_n1_5_3_0.txt │ ├── ToyData_PD_n1_5_3_1.txt │ ├── ToyData_PD_n1_5_4_0.txt │ ├── ToyData_PD_n1_5_4_1.txt │ ├── ToyData_PD_n1_5_5_0.txt │ ├── ToyData_PD_n1_5_5_1.txt │ ├── ToyData_PD_n1_5_6_0.txt │ ├── ToyData_PD_n1_5_6_1.txt │ ├── ToyData_PD_n1_6_1_0.txt │ ├── ToyData_PD_n1_6_1_1.txt │ ├── ToyData_PD_n1_6_2_0.txt │ ├── ToyData_PD_n1_6_2_1.txt │ ├── ToyData_PD_n1_6_3_0.txt │ ├── ToyData_PD_n1_6_3_1.txt │ ├── ToyData_PD_n1_6_4_0.txt │ ├── ToyData_PD_n1_6_4_1.txt │ ├── ToyData_PD_n1_6_5_0.txt │ ├── ToyData_PD_n1_6_5_1.txt │ ├── ToyData_PD_n1_6_6_0.txt │ ├── ToyData_PD_n1_6_6_1.txt │ ├── ToyData_PD_n1_7_1_0.txt │ ├── ToyData_PD_n1_7_1_1.txt │ ├── ToyData_PD_n1_7_2_0.txt │ ├── ToyData_PD_n1_7_2_1.txt │ ├── ToyData_PD_n1_7_3_0.txt │ ├── ToyData_PD_n1_7_3_1.txt │ ├── ToyData_PD_n1_7_4_0.txt │ ├── ToyData_PD_n1_7_4_1.txt │ ├── ToyData_PD_n1_7_5_0.txt │ ├── ToyData_PD_n1_7_5_1.txt │ ├── ToyData_PD_n1_7_6_0.txt │ ├── ToyData_PD_n1_7_6_1.txt │ ├── ToyData_PD_n1_8_1_0.txt │ ├── ToyData_PD_n1_8_1_1.txt │ ├── ToyData_PD_n1_8_2_0.txt │ ├── ToyData_PD_n1_8_2_1.txt │ ├── ToyData_PD_n1_8_3_0.txt │ ├── ToyData_PD_n1_8_3_1.txt │ ├── ToyData_PD_n1_8_4_0.txt │ ├── ToyData_PD_n1_8_4_1.txt │ ├── ToyData_PD_n1_8_5_0.txt │ ├── ToyData_PD_n1_8_5_1.txt │ ├── ToyData_PD_n1_8_6_0.txt │ ├── ToyData_PD_n1_8_6_1.txt │ ├── ToyData_PD_n1_9_1_0.txt │ ├── ToyData_PD_n1_9_1_1.txt │ ├── ToyData_PD_n1_9_2_0.txt │ ├── ToyData_PD_n1_9_2_1.txt │ ├── ToyData_PD_n1_9_3_0.txt │ ├── ToyData_PD_n1_9_3_1.txt │ ├── ToyData_PD_n1_9_4_0.txt │ ├── ToyData_PD_n1_9_4_1.txt │ ├── ToyData_PD_n1_9_5_0.txt │ ├── ToyData_PD_n1_9_5_1.txt │ ├── ToyData_PD_n1_9_6_0.txt │ └── ToyData_PD_n1_9_6_1.txt ├── generate_shape_data.m ├── printToyDataPDtoTextFiles.m ├── randi_helper.m └── torusreject.m └── vecs_from_PIs.m /README.md: -------------------------------------------------------------------------------- 1 | # PersistenceImages 2 | This code accompanies the paper "Persistence Images: A Stable Vector Representation of Persistent Homology" by Henry Adams, Sofya Chepushtanova, Tegan Emerson, Eric Hanson, Michael Kirby, Francis Motta, Rachel Neville, Chris Peterson, Patrick Shipman, and Lori Ziegelmeier, Journal of Machine Learning Research 18 (2017), Number 8, 1-35. The code is mainly written for use with Matlab. 3 | 4 | Three different implementations of persistence images in Python (instead of Matlab) are available at https://gitlab.com/csu-tda/PersistenceImages, at https://github.com/sauln/persim, and at https://github.com/scikit-tda/persim. 5 | A fourth implementation in Julia is available at https://github.com/chronchi/PersistenceImage.jl. 6 | -------------------------------------------------------------------------------- /git-tips.txt: -------------------------------------------------------------------------------- 1 | Before any of the following will work, you probably have to first download git (available at https://git-scm.com) on your local machine. 2 | 3 | Open the terminal on your computer. To create a local version on your computer of the folder PersistenceImages (with the latest version of our GitHub repository code), change directory to where you want this folder to go. Enter the command 4 | git clone https://github.com/CSU-TDA/PersistenceImages.git 5 | Change directories to this folder. 6 | cd PersistenceImages 7 | To see the log of changes, type 8 | git log 9 | Press "Enter" to see more changes, and press "q" to exit. 10 | 11 | You only need to clone the folder PersistenceImages once. In the future, just change directories to the folder PersistenceImages on your computer and type 12 | git pull 13 | to pull the latest code from the online repository. Remember to run the command "git pull" every time before you make edits (to ensure that you are only editing the latest version). 14 | 15 | Modify any files you like in the folder PersistentImages. Type 16 | git status 17 | to see which files you have changed. If you have added any new files, then you must type 18 | git add [filename] 19 | to tell git to start paying attention to this file. If you have removed any files and no longer want git to keep track of them, type 20 | git rm [filename] 21 | Once you are ready to commit your changes to the online repository, first type 22 | git pull 23 | to make sure nobody else has made changes in the meantime. This will not overwrite the local changes you just made -- it may ask you to resolve conflicts if others have been editing the online repository in the meantime. Type 24 | git commit -m "enter message describing commit here" 25 | to commit your changes to your local repository. Describe the changes you made inside the quotes above. This commits your changes only to your local repository on your computer. Type 26 | git push 27 | to push these changes to the online repository. You can now go to the website 28 | https://github.com/CSU-TDA/PersistenceImages.git 29 | to double-check that your changes have been made successfully. -------------------------------------------------------------------------------- /matlab_code/bump.m: -------------------------------------------------------------------------------- 1 | function [b] = bump(b_p_data, params) 2 | % bump determines the approximate value at radius of a smooth, 3 | % rotationally-invariant bump function h(x) with the properties: 4 | % 5 | % 1.) h(z) = 1 if |z| <= minradius 6 | % 2.) 0 < h(z) < 1 if minradius < |z| < maxradius 7 | % 3.) h(z) = 0 if maxradius < |z| 8 | 9 | % persistence length 10 | radius = b_p_data(2); 11 | 12 | % Parameters controlling range of non-zero, non-trivial weightings 13 | maxradius = params(1); 14 | minradius = params(2); 15 | 16 | % Parameters controlling bump function near maxradius and minradius 17 | M = [params(3),params(4)]; 18 | 19 | % Parameter to control error approximation. 20 | resolution = params(5); 21 | 22 | if radius > maxradius 23 | b = 1; 24 | return; 25 | elseif radius < minradius 26 | b = 0; 27 | return; 28 | end 29 | 30 | 31 | x = linspace(minradius, maxradius, resolution); 32 | g = zeros(1,resolution); 33 | 34 | for i=1:resolution 35 | g(i) = f(x(i)-minradius, M(1))*f(maxradius-x(i), M(2)); 36 | end 37 | 38 | b = h(x,g,radius); 39 | 40 | 41 | 42 | % ================ 43 | % Local functions 44 | % ================ 45 | % Smoothly varying, monotonic function whose range is [0,1). 46 | function y = f(x,M) 47 | if x > 0 48 | y = exp(-M/x); 49 | else 50 | y = 0; 51 | end 52 | 53 | 54 | 55 | % Integral function h(x) = int_{-inf}^{radius} g(x) / int_{-inf}^{inf} g(x) 56 | function z = h(x,g,radius) 57 | [~,idx] = min(abs(x-radius)); 58 | if idx < 2 59 | z = 0; 60 | else 61 | z = trapz(x(1:idx),g(1:idx))/trapz(x,g); 62 | end 63 | 64 | 65 | -------------------------------------------------------------------------------- /matlab_code/circleAndTorusIntervals.mat: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/CSU-TDA/PersistenceImages/4a7ebfc435494e2c5467d4e98d8ecb67b0614bf3/matlab_code/circleAndTorusIntervals.mat -------------------------------------------------------------------------------- /matlab_code/demo.m: -------------------------------------------------------------------------------- 1 | %% Example 0: Generate persistence images from a "grid-like" persistence diagram with a linear weighting function 2 | 3 | HKs=cell(1,1); 4 | HKs{1,1}=[0,0;0,1;0,2;0,3;0,4;1,1;1,2;1,3;1,4;2,2;2,3;2,4;3,3;3,4;4,4]; 5 | res=300; 6 | sig=0.01; 7 | weight_func=@linear_ramp; 8 | params=[0.05,3.5]; 9 | %use default setting for hard/soft bounds or specify type=0 or type=1 10 | [ PIs ] = make_PIs(HKs, res, sig, weight_func, params); 11 | 12 | % Plot a persistence diagram and its persistence image. 13 | HK1=HKs{1,1}; 14 | plot(HK1(:,1),HK1(:,2),'o') 15 | hold on; 16 | plot([0,4],[0,4]) % plot the diagonal line 17 | figure 18 | imagesc(PIs{1,1}) 19 | 20 | %% Example 1: Generate persistence images with linear weighting function 21 | 22 | HKs=cell(10,5); 23 | for i=1:10 24 | for j=1:5 25 | b=rand(50,1); 26 | HKs{i,j}=[b, b+(2/3)*rand(50,1)]; 27 | end 28 | end 29 | res=20; 30 | sig=0.001; 31 | weight_func=@linear_ramp; 32 | params=[0.05,0.5]; 33 | % use default setting for hard/soft bounds or specify type=0 or type=1 34 | [ PIs ] = make_PIs(HKs, res, sig, weight_func, params); 35 | 36 | % Plot a persistence diagram and its persistence image. 37 | HK1=HKs{1,1}; 38 | plot(HK1(:,1),HK1(:,2),'o') 39 | hold on; 40 | plot([0,1.5],[0,1.5]) % plot the diagonal line 41 | figure 42 | imagesc(PIs{1,1}) 43 | 44 | %% Example 2: Generate persistence images with bump weighting function 45 | HKs=cell(10,5); 46 | for i=1:10 47 | for j=1:5 48 | b=rand(50,1); 49 | HKs{i,j}=[b, b+(2/3)*rand(50,1)]; 50 | end 51 | end 52 | res=20; 53 | sig=.01; 54 | 55 | maxradius = 1.5; 56 | minradius = .5; 57 | maxgrow = 1; 58 | mingrow = 1; 59 | resolution = 10000; 60 | 61 | weight_func = @bump; 62 | params = [maxradius, minradius, maxgrow, mingrow, resolution]; 63 | 64 | %use default setting for hard/soft bounds or specify type=0 or type=1 65 | [ PIs ] = make_PIs(HKs, res, sig, weight_func, params); 66 | 67 | % Plot a persistence diagram and its persistence image. 68 | HK1=HKs{1,1}; 69 | plot(HK1(:,1),HK1(:,2),'o') 70 | hold on; 71 | plot([0,1.5],[0,1.5]) % plot the diagonal line 72 | figure 73 | imagesc(PIs{1,1}) 74 | 75 | %% Example 3: Generate PVs with bump weighting function 76 | HKs=cell(10,5); 77 | for i=1:10 78 | for j=1:5 79 | HKs{i,j}=[zeros(50,1),(2/3)*rand(50,1)]; 80 | end 81 | end 82 | res=20; 83 | sig=.01; 84 | 85 | maxradius = 1.5; 86 | minradius = .5; 87 | maxgrow = 1; 88 | mingrow = 1; 89 | resolution = 10000; 90 | 91 | weight_func = @bump; 92 | params = [maxradius, minradius, maxgrow, mingrow, resolution]; 93 | 94 | % use default setting for hard/soft bounds or specify type=0 or type=1 95 | [ PIs ] = make_PVs(HKs, res, sig, weight_func, params); 96 | 97 | % Plot a persistence diagram and its persistence image. 98 | HK1=HKs{1,1}; 99 | plot(HK1(:,1),HK1(:,2),'o') 100 | hold on; 101 | plot([0,1.5],[0,1.5]) % plot the diagonal line 102 | figure 103 | imagesc(PIs{1,1}) 104 | 105 | %% Example 4: Making persistence images with the default settings 106 | % the default settings are a linear weighting function over the interval [0, 107 | % max_persistence], resolution=25, sigma=1/2*(pixel height), and hard 108 | % boundaries. 109 | 110 | HKs=cell(10,5); 111 | for i=1:10 112 | for j=1:5 113 | b=rand(50,1); 114 | HKs{i,j}=[b,b+(2/3)*rand(50,1)]; 115 | end 116 | end 117 | [ PIs ] = make_PIs(HKs); 118 | 119 | % Plot a persistence diagram and its persistence image. 120 | HK1=HKs{1,1}; 121 | plot(HK1(:,1),HK1(:,2),'o') 122 | hold on; 123 | plot([0,1.5],[0,1.5]) % plot the diagonal line 124 | figure 125 | imagesc(PIs{1,1}) 126 | -------------------------------------------------------------------------------- /matlab_code/flexible_weighting_function.m: -------------------------------------------------------------------------------- 1 | % Random persistence diagram 2 | intervals = zeros(500,2); 3 | intervals(:,1) = rand(500,1); 4 | intervals(:,2) = intervals(:,1) + 2/3*abs(randn(500,1)); 5 | 6 | % ================= 7 | % Example 1 8 | % ================= 9 | % Specifying the function. In this case it is a piecewise linear function. 10 | % which is identically 0 until params(1) and identically 1 after params(2). 11 | weight_func = @linear_ramp; % <-- the user would input the function handle specifying their weighting function 12 | params = [1/5,1.5]; % <-- the user would supply any parameters needed to specify their weighting function. 13 | 14 | % ------------------------------------------------------------------------------------ 15 | % fixed code that computes weight at each persistence interval 16 | weights = arrayfun(@(row) weight_func(intervals(row,:), params), 1:size(intervals,1))'; 17 | % ------------------------------------------------------------------------------------ 18 | 19 | % plot the persistence diagram with points colored by weighting function 20 | figure 21 | scatter(intervals(:,1),intervals(:,2),[],weights,'filled') 22 | axis equal 23 | axis([0,1,0,max(intervals(:,2))]) 24 | colorbar 25 | 26 | 27 | % ================= 28 | % Example 2 29 | % ================= 30 | % Specifying the function. In this case it is our bump function with 31 | % a host of parameters specifying its form. 32 | 33 | maxradius = 1.5; 34 | minradius = .5; 35 | maxgrow = 1; 36 | mingrow = 1; 37 | resolution = 10000; 38 | 39 | weight_func = @bump; % <-- the user would input the function handle specifying their weighting function 40 | params = [maxradius, minradius, maxgrow, mingrow, resolution]; 41 | % <-- the user would supply any parameters needed to specify their weighting function. 42 | 43 | % ------------------------------------------------------------------------------------ 44 | % fixed code that computes weight at each persistence interval 45 | weights = arrayfun(@(row) weight_func(intervals(row,:), params), 1:size(intervals,1))'; 46 | % ------------------------------------------------------------------------------------ 47 | 48 | % plot the persistence diagram with points colored by weighting function 49 | figure 50 | scatter(intervals(:,1),intervals(:,2),[],weights,'filled') 51 | axis equal 52 | axis([0,1,0,max(intervals(:,2))]) 53 | colorbar 54 | 55 | 56 | % ================= 57 | % Example 3 58 | % ================= 59 | % Compare the two methods of computing the linear weighting function 60 | weight_func = @linear_ramp; 61 | params = [1/5,1.5]; 62 | tic 63 | linear_ramp_weights = arrayfun(@(row) weight_func(intervals(row,:), params), 1:size(intervals,1))'; 64 | linear_ramp_time = toc 65 | 66 | 67 | weight_func = @bump; 68 | params = [1.5, 1/5, 0, 0, 10000]; 69 | tic 70 | linear_bump_function_weights = arrayfun(@(row) weight_func(intervals(row,:), params), 1:size(intervals,1))'; 71 | linear_bump_function_time = toc 72 | 73 | max_weighting_function_difference = max(abs(linear_bump_function_weights - linear_ramp_weights)) 74 | 75 | 76 | % ================= 77 | % Example 4 78 | % ================= 79 | % Specifying the function. In this case it is a function which is 0 80 | % outside a rectangular region of the persistence diagram. 81 | 82 | 83 | weight_func = @rect_region; % <-- the user would input the function handle specifying their weighting function 84 | params = [0,1/2,1,1.5];% <-- the user would supply any parameters needed to specify their weighting function. 85 | 86 | % ------------------------------------------------------------------------------------ 87 | % fixed code that computes weight at each persistence interval 88 | weights = arrayfun(@(row) weight_func(intervals(row,:), params), 1:size(intervals,1))'; 89 | % ------------------------------------------------------------------------------------ 90 | 91 | % plot the persistence diagram with points colored by weighting function 92 | figure 93 | scatter(intervals(:,1),intervals(:,2),[],weights,'filled') 94 | axis equal 95 | axis([0,1,0,max(intervals(:,2))]) 96 | colorbar -------------------------------------------------------------------------------- /matlab_code/linear_ramp.m: -------------------------------------------------------------------------------- 1 | function [y] = linear_ramp(b_p_data, params) 2 | 3 | 4 | min_radius = params(1); 5 | max_radius = params(2); 6 | 7 | x = b_p_data(:,2); 8 | 9 | if x <= min_radius 10 | y = 0; 11 | elseif x >= max_radius 12 | y = 1; 13 | else 14 | y = (x - min_radius)/(max_radius - min_radius); 15 | end 16 | 17 | 18 | 19 | end 20 | 21 | -------------------------------------------------------------------------------- /matlab_code/minimal_example.m: -------------------------------------------------------------------------------- 1 | %% A minimal example using the make_PIs and make_PVs functions 2 | close all 3 | clear all 4 | load('circleAndTorusIntervals.mat') 5 | % circleAndTorusIntervals_H1 is a 1x2 cell containing the H1 intervals for 6 | % points sampled from a noisy circle and torus. 7 | % circleAndTorusIntervals_H0 is a 1x2 cell containing the H0 intervals from 8 | % points sampled from a noisy circle and torus. 9 | 10 | sig=0.0001; 11 | res=50; 12 | [H1_PIs] = make_PIs(circleAndTorusIntervals_H1, res, sig); 13 | figure, imagesc(H1_PIs{1}) 14 | title('Persistant Image for H1 diagram of points sampled from a noisy circle') 15 | figure, imagesc(H1_PIs{2}) 16 | title('Persistant Image for H1 diagram of points sampled from a noisy torus') 17 | 18 | % Since all of the H0 intervals begin at time zero, we produce a 19 | % 1-dimensional "vector grid" on top of this persistence diagram (instead 20 | % of a 2-dimesional "image grid"). 21 | [H0_PVs] = make_PVs(circleAndTorusIntervals_H0, res, sig); 22 | figure, imagesc(H0_PVs{1}), axis equal 23 | title('Persistant Vector for H0 diagram of points sampled from a noisy circle') 24 | figure, imagesc(H0_PVs{2}), axis equal 25 | title('Persistant Vector for H0 diagram of points sampled from a noisy torus') 26 | 27 | %% Convert images to vectors 28 | 29 | % Before applying machine learning techniques, it is often helpful to 30 | % conver a persistent image in matrix format into a vector by concatenating 31 | % columns. This is accomplished by the following command. 32 | H1_vecs = vecs_from_PIs(H1_PIs); -------------------------------------------------------------------------------- /matlab_code/rect_region.m: -------------------------------------------------------------------------------- 1 | function [y] = rect_region(interval,params) 2 | 3 | y = 1; 4 | if interval(1) < params(1) || interval(1) > params(2) || interval(2) < params(3) || interval(2) > params(4) 5 | y = 0; 6 | return 7 | end 8 | 9 | 10 | end 11 | 12 | -------------------------------------------------------------------------------- /matlab_code/sixShapeClasses/ToyData_PD_TextFiles/README.md: -------------------------------------------------------------------------------- 1 | # 6 shape classes 2 | This subfolder contains persistence diagram data for 6 shape classes: 3 | (1) A unit cube 4 | (2) A circle of diameter one 5 | (3) A sphere of diameter one 6 | (4) Three clusters with centers randomly chosen in the unit cube 7 | (5) Three clusters within three clusters 8 | (6) A torus with a major diameter of one and a minor diameter of one half. 9 | These shape classes are described in Section 6.1 of the paper "Persistence Images: A Stable Vector Representation of Persistent Homology" 10 | 11 | We produce 25 point clouds of 500 randomly sampled points from each shape class. We then add a level of Gaussian noise to each point, at a noise level neta=0.1 or neta=0.05. We then have already computed the persistent homology intervals in homological dimension i=0 and i=1. 12 | 13 | For example, the file 14 | ToyData_PD_n05_23_6_0.txt 15 | corresponds to noise level neta=0.05, the 23rd point cloud randomly sampled from shape class (6), with persistent homology computed in dimension 0. 16 | Each row of this file has two entries: the birth and death time of a 0-dimensional persistent homology interval. 17 | 18 | By contrast, the file 19 | ToyData_PD_n1_21_3_1.txt 20 | corresponds to noise level neta=1, the 21st point cloud randomly sampled from shape class (3), with persistent homology computed in dimension 1. 21 | Each row of this file has two entries: the birth and death time of a 1-dimensional persistent homology interval. 22 | 23 | Your task is to use machine learning to distinguish these six classes from each other. In a K-medoids clustering test, some accuracies and computation times are displayed for bottleneck distances, Wasserstein distances, persistence landscapes, and persistence images in Table 1 of "Persistence Images: A Stable Vector Representation of Persistent Homology". Do you have ideas for beating these accuracies or computation times? 24 | -------------------------------------------------------------------------------- /matlab_code/sixShapeClasses/ToyData_PD_TextFiles/ToyData_PD_n05_10_2_1.txt: -------------------------------------------------------------------------------- 1 | 0.01948416075214823989 0.01979640967396104068 2 | 0.02387862712664386156 0.02459203878552512074 3 | 0.02546861519148301001 0.02700320956071801393 4 | 0.02384870660179845211 0.02786135035153804868 5 | 0.02719884404118429883 0.02870881176628865827 6 | 0.02914271558507823545 0.02947520303385991677 7 | 0.02853190415045738990 0.03046230958944595030 8 | 0.02904020957767292771 0.03136384930167451790 9 | 0.02619436563502263587 0.03160635737183528637 10 | 0.02941926167642771456 0.03248139990887272122 11 | 0.03195981393265033554 0.03316523879926636803 12 | 0.03240044177903137618 0.03442616122210994528 13 | 0.03418437419813789152 0.03489016921371988766 14 | 0.03181258394429766628 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toy data to use in classification experiments. See 3 | % the paper "Persistence Images: A Stable Vector Representation of 4 | % Persistent Homology" 5 | 6 | % n is the number of points in each data set 7 | % m is the number of examples of each type 8 | % V is the variance of noise 9 | 10 | ToyData=cell(m+1,6); 11 | % Column 1: Random Point cloud (in R^3) 12 | % Column 2: circle 13 | % Column 3: sphere 14 | % Column 4: 3 clusters 15 | % Column 5: 3 clusters each with 3 smaller clusters 16 | % Column 6: torus 17 | 18 | % Random points drawn from a uniform distribution 19 | ToyData{1,1}=['Random Cloud']; 20 | for i=2:m+1 21 | X=rand(3,n); 22 | ToyData{i,1}=X; 23 | end 24 | 25 | % Generate point cloud data randomly sampled from a circle in R^2 centered 26 | % at (.5,.5) radius .5 27 | ToyData{1,2}=['Circle']; 28 | for i=2:m+1 29 | X=randn(2,n); % Guassian distribution gives even random distrubution over sphere 30 | s2=sum(X.^2,1); 31 | S=repmat(sqrt(s2),2,1); 32 | X=X./(2*S); % normamlize so that the vectors are all length 1/2 33 | X=X+repmat(.5,2,n); % shift sphere from origin to (.5,.5) 34 | Noise=V*randn(2,n); % generate noise, normal distribution, variance V 35 | X=X+Noise; 36 | %scatter(X(1,:), X(2,:)) 37 | ToyData{i,2}=X; 38 | end 39 | 40 | % Generate point cloud randomly sampled from a sphere, radius 1/2, centered 41 | % at (.5,.5,.5) with noise added 42 | ToyData{1,3}=['Sphere']; 43 | for i=2:m+1 44 | X=randn(3,n); % Guassian distribution gives even random distrubution over sphere 45 | s2=sum(X.^2,1); 46 | S=repmat(sqrt(s2),3,1); 47 | X=X./(2*S); % normamlize so that the vectors are all length 1/2 48 | X=X+repmat(.5,3,n); % shift sphere from origin to (.5,.5,.5) 49 | Noise=V*randn(3,n); % generate noise, normal distribution, variance V 50 | X=X+Noise; 51 | %scatter3(X(1,:), X(2,:),X(3,:)); 52 | ToyData{i,3}=X; 53 | end 54 | 55 | 56 | % clusters of points around three randomly chosen points in [0,1]x[0,1]x[0,1] 57 | % Note: random number of points in each cluster, at most half of all points 58 | % in a single cluster, may not be cleanly separated 59 | 60 | % 3 tight clusters of points in 3 larger clusters around same centers as first set of clusters 61 | 62 | ToyData{1,4}=['Clusters']; 63 | ToyData{1,5}=['Clusters within Clusters']; 64 | 65 | for i=2:m+1 66 | Centers=rand(3); 67 | a=max(randi_helper(floor(.45*n)),floor(.1*n)); % random number of points for first cluster, can be at most 45% of all points, but is at least 10% 68 | b=max(randi_helper(floor(.45*(n-a))), floor( (n-a)*.1)); % random number of pts in 2nd cluster, at least 10% remaining, at most 45% 69 | c=n-a-b; 70 | X1=cat(2,repmat(Centers(:,1),1,a),repmat(Centers(:,2),1,b),repmat(Centers(:,3),1,c)); 71 | Y=.05*randn(3,n); 72 | X1=X1+Y; 73 | %scatter3(X1(1,:), X1(2,:),X1(3,:)); 74 | ToyData{i,4}=X1; 75 | 76 | Centers=cat(2,repmat(Centers(:,1),1,3),repmat(Centers(:,2),1,3),repmat(Centers(:,3),1,3)); 77 | centers=.05*randn(3,9); %small cluster centers: perturbation from large cluster center 78 | Centers=Centers+centers; %center of cluster + perturbation 79 | 80 | % random number of points for each small cluster, can be at most 45% and at least 10% 81 | a1=max(randi_helper(floor(.45*a)),floor(.1*a)); 82 | a2=max(randi_helper(floor(.45*(a-a1))),floor((a-a1)*.1)); 83 | a3=a-a1-a2; 84 | 85 | X=cat(2,repmat(Centers(:,1),1,a1),repmat(Centers(:,2),1,a2),repmat(Centers(:,3),1,a3)); 86 | 87 | b1=max(randi_helper(floor(.45*b)),floor(b*.1)); % random number of points for first small cluster, can be at most 45% of all points and at least 10% 88 | b2=max(randi_helper(floor(.45*(b-b1))),floor((b-b1)*.1)); 89 | b3=b-b1-b2; 90 | 91 | Y=cat(2,repmat(Centers(:,4),1,b1),repmat(Centers(:,5),1,b2),repmat(Centers(:,6),1,b3)); 92 | 93 | c1=max(randi_helper(floor(.45*c)),floor(c*.1)); %random number of points for first small cluster, can be at most 45% of all points and at least 10% 94 | c2=max(randi_helper(floor(.45*(c-c1))), floor((c-c1)*.1)); 95 | c3=c-c1-c2; 96 | 97 | Z=cat(2,repmat(Centers(:,7),1,c1),repmat(Centers(:,8),1,c2),repmat(Centers(:,9),1,c3)); 98 | 99 | X=cat(2,X,Y,Z); 100 | N=.02*randn(3,length(X)); %randomly disperse from centers 101 | X=X+N; 102 | %scatter3(X(1,:), X(2,:),X(3,:)); 103 | ToyData{i,5}=X; 104 | 105 | end 106 | 107 | % Generate point cloud data randomly sampled from a torus in R^3 cantered 108 | % at (.5,.5,.5) R=.5, r=.25 109 | ToyData{1,6}=['Torus']; 110 | for i=2:m+1 111 | [theta, phi]=torusreject(n); 112 | X=[(.35+.15.*cos(theta)).*cos(phi), (.35+.15.*cos(theta)).*sin(phi),sin(theta)]; 113 | X=X'+repmat(.5,3,n); % shift sphere from origin to (.5,.5,.5) 114 | Noise=V*randn(3,n); % generate noise, normal distribution, variance V 115 | X=X+Noise; 116 | %scatter3(X(1,:), X(2,:),X(3,:)); 117 | ToyData{i,6}=X; 118 | end 119 | 120 | 121 | end 122 | %save('toydata_n1','toydata_n1'); 123 | -------------------------------------------------------------------------------- /matlab_code/sixShapeClasses/printToyDataPDtoTextFiles.m: -------------------------------------------------------------------------------- 1 | % This is a Matlab script for printing the 6 classes of shape data, generated for example using the command generate_shape_data(n,m,V), to individual text files (on text file for each random sample of points.) 2 | 3 | load ToyData_PD_n05.mat 4 | % The cell array ToyData_barcode_n05 is of size 26 x 6 x 2: 25 instances of 5 | % each shape (plus an extra row with shape names at the top), 6 shape 6 | % classes, and 2 homological dimensions (0 and 1). 7 | for i=2:26 8 | for j=1:6 9 | for k=1:2 10 | % We reindix the filenames to get shapes numbered 1-25 and homological dimension numbered 0-1. 11 | fname = strcat('ToyData_PD_TextFiles/ToyData_PD_n05_',int2str(i-1),'_',int2str(j),'_',int2str(k-1),'.txt'); 12 | fileID = fopen(fname,'w'); 13 | fprintf(fileID,'%.20f %.20f\n',ToyData_barcode_n05{i,j,k}'); 14 | fclose(fileID); 15 | end 16 | end 17 | end 18 | 19 | load ToyData_PD_n1.mat 20 | for i=2:26 21 | for j=1:6 22 | for k=1:2 23 | % We reindix the filenames to get shapes numbered 1-25 and homological dimension numbered 0-1. 24 | fname = strcat('ToyData_PD_TextFiles/ToyData_PD_n1_',int2str(i-1),'_',int2str(j),'_',int2str(k-1),'.txt'); 25 | fileID = fopen(fname,'w'); 26 | fprintf(fileID,'%.20f %.20f\n',ToyData_barcode_n1{i,j,k}'); 27 | fclose(fileID); 28 | end 29 | end 30 | end 31 | -------------------------------------------------------------------------------- /matlab_code/sixShapeClasses/randi_helper.m: -------------------------------------------------------------------------------- 1 | function integer=randi_helper(k) 2 | 3 | % This function returns randi(k) if integer k is at least one, and 4 | % otherwise it returns 0. The reason is that randi(0) gives an error in 5 | % Matlab, and this function avoids that! 6 | 7 | if k>=1 8 | integer=randi(k); 9 | else 10 | integer=0; 11 | end -------------------------------------------------------------------------------- /matlab_code/sixShapeClasses/torusreject.m: -------------------------------------------------------------------------------- 1 | function [ theta, phi ] = torusreject(n) 2 | 3 | % Generate sampled angles [0,2PI) 4 | % From method detailed in Sampling From A Manifold by Persi Diaconis, Susan 5 | % Holmes, and Mehrdad Shahshahani 6 | % R=.5 r=.25 7 | 8 | x=2*pi*rand(4*n,1); 9 | y=(1/pi)*rand(4*n,1); 10 | fx=(1+(.25/.5)*cos(x))/(2*pi); 11 | 12 | theta=x(y